What are the applications of ant colony optimization algorithms?
The first ant colony optimisation algorithm was introduced by Marco Dorigo in the report Positive Feedback as a Search Strategy (1991) and his PhD thesis Optimization, Learning and Natural Algorithms (1992). He's still one of the leading figures in the field of swarm intelligence (having also written or co-written several papers and books). Another important person that contributed to ACO algorithms is Luca Gambardella (co-director of IDSIA).

There are several ACO algorithms. They are all based on the way real ants behave, that is, by leaving a substance called "pheromone" on the ground in order to communicate. More specifically, the amount of pheromone is associated with value (e.g. food): more pheromone means more value. (It should now be clear the reason behind the queues real ants form).

A list of ACO algorithms can be found at http://iridia.ulb.ac.be/~mdorigo/ACO/publications.html. For reproducibility, here's a (non-exhaustive) list:


Ant System
Elitist Ant System
Ant-Q
Ant Colony System
Max-Min Ant System
Rank-based Ant System
ANTS
Hyper Cube - ACO


ACO algorithms have been applied to combinatorial and NP-complete (e.g. the travelling salesman problem) problems. ACO algorithms are thus a collection of meta-heuristic and probabilistic algorithms (in the same family of simulated annealing) to tackle often considered intractable problems. The related Wikipedia article contains a more exhaustive section dedicated to the applications of these algorithms. ACO algorithms are often combined with local search algorithms (like the 2-opt or 3-opt).

I would suggest you to start with the travelling salesman problem, which was the first application of these algorithms. You can have a look at the reference implementations at http://iridia.ulb.ac.be/~mdorigo/ACO/aco-code/public-software.html, where you can also find software to solve specific tasks (not just the TSP, such as maximum clique problems).